Asymptotic Method of Solution for a Problem of Construction of Optimal Gas-Lift Process Modes

Mathematical model in oil extraction by gas-lift method for the case when the reciprocal value of well’s depth represents a small parameter is considered. Problem of optimal mode construction i.e., construction of optimal program trajectories and controls is reduced to the linear-quadratic optimal control problem with a small parameter. Analytic formulae for determining the solutions at the first-order approximation with respect to the small parameter are obtained. Comparison of the obtained results with known ones on a specific example is provided, which makes it, in particular, possible to use obtained results in realizations of oil extraction problems by gas-lift method.


Introduction
It is known 1-3 that the gas-lift technique of exploitation of oil wells is widely used when the gushing method does not work for the reason of insufficiency in pressure.The essence of the gas-lift method consists of the fact that by the mean of energy of injected underground gas it is possible to lift fluid to the surface.
While the gas-lift method is widely used in oil extraction for a sufficiently long period of time, construction of an adequate mathematical model is rather an actual problem.Mathematical model describing the oil lifting process in pump-compressor tubes is described in 4 .
In 4, 5 , using this model, an optimal control problem for gas-lift process is formulated, where the pressure and volume of the injected gas is used as a control parameter.In the same papers, the method of straight lines is used to reduce the optimal control problem

Mathematical Formulation of the Problem
As in 4 , mathematical model of gas-fluid mixture flow in pipes is described by the system of hyperbolic-type partial differential equations: which for x z/2L and ε 1/2L can be written as with appropriate boundary and initial conditions

Mathematical Problems in Engineering 3
It is required to determine a control minimizing the functional where P is pressure, Q is gas-fluid mixture volume and Q deb is the desired yield.
Applying the so-called method of straight lines to 2.2 and taking l 1/N, we obtain

2.6
Note that for k N 1 2.6 can be written as where Q pl , P pl denote gas-fluid outlay yield and pressure at the bottom of a well, respectively.As in 10 , a linear-quadratic optimal control problem is formulated for this system.It is required to find x, u satisfying the equation with initial condition x 0 x 0 2.9 such that the value of the functional is minimized.

Mathematical Problems in Engineering
Here, Let R Rε.Then the corresponding Euler-Lagrange control problem can be written as

2.13
Thus, initial problem 2.2 -2.5 is reduced to finding the solution of problem 2.12 , where ε is a small parameter.Therefore, the asymptotic method see 9 allowing to expand the solution of system 2.12 with respect to small parameter ε can be applied.

Application of Asymptotic Method
Let us apply the asymptotic method to the Euler-Lagrange equation 2.12 with boundary conditions for x 0 and λ T .According to 12 ,

3.1
Further, let us expand the expression 3.2 from 3.1 with respect to ε.If we denote then according to 12 , the expansion of expression 3.2 with respect to ε can be represented as: Denote the integral in 3.4 by L 0 .Then it is not difficult to show that matrix L 0 is a solution of the following Sylvester's equation: Therefore, for expression 3.2 we obtain the expansion e H 1 T εH 2 T ≈ e H 1 T εL 0 .

3.6
Introducing notations Hence, adding boundary conditions from 2.12 , we arrive to the following system of algebraic equations: which in the matrix form can be written as Multiplying 3.10 by Hence, if we denote the coefficient matrix in 3.11 by M, then it can be written as where Hence, using the fact that we obtain the inverse matrix and, consequently, multiplying the both sides of 3.11 by M −1 on the left, we obtain the following analytic formulae to determine values of x T , λ 0 , λ T : 3.17  Further, as in 3.1 , using determined in 3.17 value λ 0 , it is possible to find x t i , λ t i in the form for every t i ∈ 0 T .Therefore, expansions for x t i and u t i with respect to ε can be obtained in the form x t i e A 0 t i ε e A 0 t i S 1 C − S 1 C L 11 x 0 − L 22 e A 0 t i Ne A 0 t i x 0 L 22 e A 0 t i Nx , λ t i −Ne A 0 T x 0 Nx ε NL 22 e A 0 t i Ne A 0 t i x 0 − NL 22 e A 0 t i Nx − NL 11 x 0 − Ne A 0 t i S 1 C − NS 1 C , u t i −R −1 B λ t i .

3.19
Here L 11 , L 22 are block matrices of matrix-solutions L i of Sylvester's equation 12, 13

Figure 2 :
Figure 2: Dependence of P 2 on t.
1 t i H 2 − H 2 e H 1 t i 3.20 for every t i .